In mathematics, the fundamental class is a homology class [''M''] associated to a connected orientable compact manifold of dimension n, which corresponds to the generator of the homology group
Hn(M,\partialM;Z)\congZ
When M is a connected orientable closed manifold of dimension n, the top homology group is infinite cyclic:
Hn(M;Z)\congZ
Z\toHn(M;Z)
If M is disconnected (but still orientable), a fundamental class is the direct sum of the fundamental classes for each connected component (corresponding to an orientation for each component).
In relation with de Rham cohomology it represents integration over M; namely for M a smooth manifold, an n-form ω can be paired with the fundamental class as
\langle\omega,[M]\rangle=\intM\omega ,
which is the integral of ω over M, and depends only on the cohomology class of ω.
If M is not orientable,
Hn(M;Z)\ncongZ
Z2
Hn(M;Z2)=Z2
Z2
Z2
This
Z2
If M is a compact orientable manifold with boundary, then the top relative homology group is again infinite cyclic
Hn(M,\partialM)\congZ
See main article: Poincaré duality. The Poincaré duality theorem relates the homology and cohomology groups of n-dimensional oriented closed manifolds: if R is a commutative ring and M is an n-dimensional R-orientable closed manifold with fundamental class [M], then for all k, the map
Hk(M;R)\toHn-k(M;R)
\alpha\mapsto[M]\frown\alpha
Using the notion of fundamental class for manifolds with boundary, we can extend Poincaré duality to that case too (see Lefschetz duality). In fact, the cap product with a fundamental class gives a stronger duality result saying that we have isomorphisms
Hq(M,A;R)\congHn-q(M,B;R)
A,B
(n-1)
\partialA=\partialB=A\capB
\partialM=A\cupB
See also Twisted Poincaré duality
In the Bruhat decomposition of the flag variety of a Lie group, the fundamental class corresponds to the top-dimension Schubert cell, or equivalently the longest element of a Coxeter group.
. Allen Hatcher. Algebraic Topology. 2002. Cambridge University Press. 9780521795401. 1st. Cambridge. English. 1867354.
. Algebraic Topology . 2002 . . 9780521795401 . 1st . Cambridge . 241–254 . English . 1867354 . Allen Hatcher.